Necessary and Sufficient conditions Notes for November 8

Sufficient conditions

If A is a sufficient condition of B, then you cannot have A without B.

For example, suppose that:

A = receiving more than 50% of the votes

B = winning the election

“Receiving more than 50% of the votes is a sufficient condition of winning the election” = the candidate who receives more than 50% of the votes wins the election.

So Kerry can claim to have won the election by virtue of meeting the sufficient condition for having won: receiving more than 50% of the votes.

Aside: those who follow elections in the US know that what counts is receiving the most votes in the Electoral College. Al Gore got the most individual votes last time, but he did not win the election because George Bush received more votes in the Electoral College.

Necessary conditions

If C is a necessary condition of D, then you cannot have D without C.

For example, suppose that:

C = receiving some votes

D = winning the election

“Receiving some votes is a necessary condition of winning the election” = no one can win the election without receiving some votes.

If Kerry claims to have won the election, Bush can refute his claim by showing that the necessary condition was not met. That is, if he can show that Kerry did not receive some votes, he would show that Kerry did not win the election.

Note: you can meet the necessary condition without meeting the sufficient condition. Both candidates will receive some votes this year (the necessary condition), but only one will receive the most votes (the sufficient condition). Only the one that meets the necessary and sufficient conditions will win the election.

Refuting proposed conditions

Often, we debate about whether proposed necessary and sufficient conditions are genuine. Here’s how to do it. Look for cases that would be ruled out if the proposed necessary or sufficient condition were true. If they’re possible, or, even better, actual, then the proposal is not correct.

So, I said that if A is a sufficient condition of B, then you cannot have A without B. That means that you should look for cases in which you can have A without B.

I also said that if C is a necessary condition of D, then you cannot have D without C. In order to challenge a proposed necessary condition, look for a D without a C.

Back and forth

You can reverse things in the following way: the second part is the other condition of the first part. Thus,

A is a sufficient condition of B = B is a necessary condition of A

C is a necessary condition of D = D is a sufficient condition of C

Explaining how this works will expose one very important thing about necessary and sufficient conditions. They aren’t causes.

1. Sufficient to necessary conditions

Suppose we grant that receiving more than 50% of the votes is a sufficient condition of winning the election. And suppose we also grant that Kerry did not win the election. It follows that Kerry did not receive more than 50% of the votes: if he had received more than 50% of the votes, he would have won the election but we have granted that did not happen.

This means that B (in this case “winning the election”) is a necessary condition of A (receiving more than 50% of the votes): you cannot have A without B or, you cannot receive more than 50% of the votes without winning the election.

2. Necessary to sufficient conditions

Conversely, winning the election is a sufficient condition of receiving some votes. If we grant that receiving some votes is a necessary condition of winning the election and that Bush won the election, it follows that Bush received some votes.

That means that D is a sufficient condition of C. You cannot win the election without receiving some votes.

So these aren’t causes then, right?

Right. Winning the election is a sufficient condition of receiving some votes. But winning the election isn’t what causes a candidate to receive votes.

In general, it’s tempting to think about necessary and sufficient conditions in terms of causes and effects but that temptation should be resisted.

Be careful

Someone can win the election without receiving more than 50% of the votes. There were three candidates running in the last Presidential election: Bush, Kerry, and Nader. The winner was the one who received more votes than the other two even if that is less than 50% of the votes.

In other words, A (receiving more than 50% of the votes) is a sufficient, but not a necessary, condition of B (winning the election).

Similarly, D (winning the election) is a sufficient, but not a necessary, condition of C (receiving some votes). You can receive some votes (C) without winning the election (D), therefore, winning the election (D) cannot be a necessary condition of receiving some votes (C).